# Tensor product of free modules

Suppose $M$ and $N$ are free $R$-module($R$ is a commutative ring). The tensor product of $M\otimes_R N$ is free $R$-module? I know for projective modules it is true. How should we build its basis?

• A basis is the most natural possible: the tensor product of bases. – user26857 May 14 '17 at 19:11

$$[\bigoplus\limits_{i} R] \otimes_R [ \bigoplus\limits_j R] \cong \bigoplus\limits_{i,j} R \otimes_R R \cong \bigoplus\limits_{i,j} R$$
• Could we say its basis come from the tensor product of basis of $M$ and $N$? – B.K-Theory May 14 '17 at 19:21